Cleanup

src/fa/lc/projective.tex

@@ -7,10 +7,11 @@
     Let $E$ be a vector space over $K \in \RC$, $\seqi{F}$ be locally convex spaces over $K$, and $\seqi{T}$ where $T_i \in \hom(E; F_i)$ for all $i \in I$, then the projective topology on $E$ is locally convex.
 \end{proposition}
 \begin{proof}
-    By \ref{definition:tvs-initial},
+    By \autoref{definition:tvs-initial},
     \[
     \mathcal{B} = \bracs{\bigcap_{j \in J}T_j^{-1}(U_j) \bigg | J \subset I \text{ finite}, U_j \in \cn_{F_j}(0)}
     \]
+
     is a fundamental system of neighbourhoods at $0$. For each $i \in I$, $U_i \in \cn_{F_i}(0)$ convex, $T^{-1}(U_i)$ is also convex. Since each $F_i$ is locally convex, $\mathcal{B}$ contains a fundamental system of neighbourhoods at $0$ consisting of only convex sets.
 \end{proof}
 
@@ -19,7 +20,7 @@
     Let $(\seqi{E}, \bracsn{T^i_j|i, j \in I, i \lesssim j})$ be a downward-directed system of locally convex spaces over $K \in \RC$, then $E = \lim_{\longleftarrow}E_i$ is locally convex.
 \end{proposition}
 \begin{proof}
-    By (U) of \ref{definition:tvs-projective-limit} and \ref{definition:tvs-initial}, $E$ is equipped with the projective topology generated by the projection maps $E \to E_i$. By \ref{proposition:lc-projective-topology}, $E$ is locally convex.
+    By (U) of \autoref{definition:tvs-projective-limit} and \autoref{definition:tvs-initial}, $E$ is equipped with the projective topology generated by the projection maps $E \to E_i$. By \autoref{proposition:lc-projective-topology}, $E$ is locally convex.
 \end{proof}
 
 \begin{proposition}[{{\cite[II.5.4]{SchaeferWolff}}}]
@@ -32,6 +33,7 @@
         \[
         \pi^U_V: E_U \to E_V \quad x + M_U \mapsto x + M_V
         \]
+
         then $\pi^U_V \in L(E_U; E_V)$.
         \item $(\bracsn{E_U}_{U \in \mathcal{B}}, \bracs{\pi^U_V|U, V \in \mathcal{B}, U \subset V})$ is a downward-directed system of topological vector spaces.
         \item The map $\pi \in L(E, \lim_{\longleftarrow}E_U)$ induced by $\bracs{\pi_U}_{U \in \mathcal{B}}$ is a bijection.
@@ -39,10 +41,11 @@
         \[
         E = \lim_{\longleftarrow}E_U = \lim_{\longleftarrow} \ol E_U
         \]
+
     \end{enumerate}
 \end{proposition}
 \begin{proof}
-    (1): Since $V \supset U$, $[\cdot]_V \ge [\cdot]_U$, so $M_V \supset M_U$. Thus $\ker(\pi_V) \supset M_U$. By (U) of the quotient (\ref{definition:tvs-quotient}), $\pi_V$ factors through $E_U$ as $\pi^U_V$, so $\pi^U_V \in L(E_U; E_V)$.
+    (1): Since $V \supset U$, $[\cdot]_V \ge [\cdot]_U$, so $M_V \supset M_U$. Thus $\ker(\pi_V) \supset M_U$. By (U) of the \hyperref[quotient]{definition:tvs-quotient}, $\pi_V$ factors through $E_U$ as $\pi^U_V$, so $\pi^U_V \in L(E_U; E_V)$.
 
     (2): Since $\mathcal{B}$ is a fundamental system of neighbourhoods, it is downward-directed under inclusion. For any $U, V, W \in \mathcal{B}$ with $U \subset V \subset W$, $M_U \supset M_V \supset M_W$. Thus $\pi^U_W = \pi^V_W \circ \pi^U_V$.
 
@@ -54,11 +57,12 @@
     \[
     \pi_W(x_U) = \pi_W^U \circ \pi_U(x_U) = \pi_W^U p_U(x)
     \]
+
     Thus for any $U' \in \mathcal{B}$ with $U \subset W$, $[x_U - x_{U'}]_W = 0$, and $x_U - x_{U'} \in W$. Therefore $\bracs{x_U}_{U \in \mathcal{B}}$ is a Cauchy net, and converges to $x_0 \in E$ by completeness of $E$.
 
     For any $U \in \mathcal{B}$, $\pi_U(x_0) = \lim_{V \in \mathcal{B}}\pi_U(x_V) = p_U(x)$, so $\pi(x_0) = x$, and $\pi$ is surjective.
 
-    (4): Since $\mathcal{B} \subset \cn_E(0)$ is a fundamental system of neighbourhoods, the topology on $E$ is the projective topology generated by $\bracs{\pi_U|U \in \mathcal{B}}$. As $\pi_U \circ \pi^{-1} = p_U \in L(\lim E_U; E_U)$ for all $U \in \mathcal{B}$, $\pi^{-1} \in L(\lim E_U; E)$ by (U) of the projective topology (\ref{definition:tvs-initial}).
+    (4): Since $\mathcal{B} \subset \cn_E(0)$ is a fundamental system of neighbourhoods, the topology on $E$ is the projective topology generated by $\bracs{\pi_U|U \in \mathcal{B}}$. As $\pi_U \circ \pi^{-1} = p_U \in L(\lim E_U; E_U)$ for all $U \in \mathcal{B}$, $\pi^{-1} \in L(\lim E_U; E)$ by (U) of the \hyperref[projective topology]{definition:tvs-initial}.
 
     Let $x \in \lim\ol{E}_U$ and $V \in \cn(x)$. Since $\mathcal{B}$ is downward-directed and $\lim\ol{E}_U$ is equipped with the projective topology induced by $\bracs{p_U|U \in \mathcal{B}}$, there exists $U \in \mathcal{B}$ and $W \in \cn_{\ol E_U}(x)$ such that $p_U^{-1}(W) \subset V$. As $E_U$ is dense in $\ol E_U$, there exists $y_U \in W$, and $y \in E$ such that $y_U = \pi_U(y)$. Therefore $\pi(y_U) \in p_U^{-1}(W) \subset V$, and $\lim E_U$ is dense in $\lim \ol{E}_U$.